How to adjust view clipping distance when using object as camera?
It turns out this is actually a feature, of sorts.

When a lamp is set as a camera, the viewport is only rendered within the lamp's shadow buffer clipping extents; that is shadow_buffer_clip_startand shadow_buffer_clip_end.

Unfortunately, these properties are not presented in the GUI for every render engine, or even for every lamp type.

In 2.7x, they are only present in the game engine, on a spot or sun lamp:



In 2.8, using Eevee, only shadow_buffer_clip_start is displayed for point, spot and area lamps. To get shadow_buffer_clip_end, the lamp type must be sun.



As far as I am aware, these properties are not exposed to the GUI at all under BI or Cycles.



While this behavior does seem to be at least intentional, it probably could use some improvement. For instance, perhaps the view extents ought only to be clipped for engines and lamp types where the shadow buffer clipping is applicable. With that in mind, I've submitted a bug report here.